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Properties

Label 2-1100-55.4-c1-0-6
Degree 22
Conductor 11001100
Sign 0.320−0.947i0.320 - 0.947i
Analytic cond. 8.783548.78354
Root an. cond. 2.963702.96370
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.151 + 0.0492i)3-s + (−1.93 + 0.628i)7-s + (−2.40 − 1.74i)9-s + (2.88 + 1.63i)11-s + (−0.384 + 0.528i)13-s + (2.52 + 3.47i)17-s + (0.919 − 2.82i)19-s − 0.324·21-s + 6.11i·23-s + (−0.559 − 0.770i)27-s + (2.63 + 8.11i)29-s + (4.34 + 3.15i)31-s + (0.357 + 0.389i)33-s + (−3.28 + 1.06i)37-s + (−0.0843 + 0.0612i)39-s + ⋯
L(s)  = 1  + (0.0875 + 0.0284i)3-s + (−0.731 + 0.237i)7-s + (−0.802 − 0.582i)9-s + (0.870 + 0.492i)11-s + (−0.106 + 0.146i)13-s + (0.612 + 0.843i)17-s + (0.210 − 0.649i)19-s − 0.0708·21-s + 1.27i·23-s + (−0.107 − 0.148i)27-s + (0.489 + 1.50i)29-s + (0.780 + 0.567i)31-s + (0.0622 + 0.0678i)33-s + (−0.540 + 0.175i)37-s + (−0.0134 + 0.00980i)39-s + ⋯

Functional equation

Λ(s)=(1100s/2 ΓC(s) L(s)=((0.320−0.947i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1100s/2 ΓC(s+1/2) L(s)=((0.320−0.947i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    22⋅52⋅112^{2} \cdot 5^{2} \cdot 11
Sign: 0.320−0.947i0.320 - 0.947i
Analytic conductor: 8.783548.78354
Root analytic conductor: 2.963702.96370
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1100(1049,⋅)\chi_{1100} (1049, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1100, ( :1/2), 0.320−0.947i)(2,\ 1100,\ (\ :1/2),\ 0.320 - 0.947i)

Particular Values

L(1)L(1) ≈\approx 1.2875353501.287535350
L(12)L(\frac12) ≈\approx 1.2875353501.287535350
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
11 1+(−2.88−1.63i)T 1 + (-2.88 - 1.63i)T
good3 1+(−0.151−0.0492i)T+(2.42+1.76i)T2 1 + (-0.151 - 0.0492i)T + (2.42 + 1.76i)T^{2}
7 1+(1.93−0.628i)T+(5.66−4.11i)T2 1 + (1.93 - 0.628i)T + (5.66 - 4.11i)T^{2}
13 1+(0.384−0.528i)T+(−4.01−12.3i)T2 1 + (0.384 - 0.528i)T + (-4.01 - 12.3i)T^{2}
17 1+(−2.52−3.47i)T+(−5.25+16.1i)T2 1 + (-2.52 - 3.47i)T + (-5.25 + 16.1i)T^{2}
19 1+(−0.919+2.82i)T+(−15.3−11.1i)T2 1 + (-0.919 + 2.82i)T + (-15.3 - 11.1i)T^{2}
23 1−6.11iT−23T2 1 - 6.11iT - 23T^{2}
29 1+(−2.63−8.11i)T+(−23.4+17.0i)T2 1 + (-2.63 - 8.11i)T + (-23.4 + 17.0i)T^{2}
31 1+(−4.34−3.15i)T+(9.57+29.4i)T2 1 + (-4.34 - 3.15i)T + (9.57 + 29.4i)T^{2}
37 1+(3.28−1.06i)T+(29.9−21.7i)T2 1 + (3.28 - 1.06i)T + (29.9 - 21.7i)T^{2}
41 1+(−1.47+4.53i)T+(−33.1−24.0i)T2 1 + (-1.47 + 4.53i)T + (-33.1 - 24.0i)T^{2}
43 1−10.1iT−43T2 1 - 10.1iT - 43T^{2}
47 1+(−6.94−2.25i)T+(38.0+27.6i)T2 1 + (-6.94 - 2.25i)T + (38.0 + 27.6i)T^{2}
53 1+(6.53−8.99i)T+(−16.3−50.4i)T2 1 + (6.53 - 8.99i)T + (-16.3 - 50.4i)T^{2}
59 1+(1.70+5.24i)T+(−47.7+34.6i)T2 1 + (1.70 + 5.24i)T + (-47.7 + 34.6i)T^{2}
61 1+(−7.77+5.64i)T+(18.8−58.0i)T2 1 + (-7.77 + 5.64i)T + (18.8 - 58.0i)T^{2}
67 1+5.60iT−67T2 1 + 5.60iT - 67T^{2}
71 1+(0.0442−0.0321i)T+(21.9−67.5i)T2 1 + (0.0442 - 0.0321i)T + (21.9 - 67.5i)T^{2}
73 1+(6.78−2.20i)T+(59.0−42.9i)T2 1 + (6.78 - 2.20i)T + (59.0 - 42.9i)T^{2}
79 1+(−2.37−1.72i)T+(24.4+75.1i)T2 1 + (-2.37 - 1.72i)T + (24.4 + 75.1i)T^{2}
83 1+(7.25+9.98i)T+(−25.6+78.9i)T2 1 + (7.25 + 9.98i)T + (-25.6 + 78.9i)T^{2}
89 1+0.00487T+89T2 1 + 0.00487T + 89T^{2}
97 1+(10.7−14.7i)T+(−29.9−92.2i)T2 1 + (10.7 - 14.7i)T + (-29.9 - 92.2i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.795586355369149215741290075620, −9.240017956252944615584756944464, −8.575943188820311522775710300553, −7.46342977767049513090816107466, −6.56700328388650369729293164589, −5.93875388231027215972275861976, −4.85386047636054773540423104868, −3.62181558279128363125791730473, −2.94785858244070016772457318457, −1.35799894403851293058557352023, 0.60622658304716839612024982453, 2.40547602522024944569772199486, 3.33821494920828442366848636194, 4.36919214863017046308314193776, 5.56081271233961343205614175715, 6.26543845457092303110864537684, 7.17342767226686509074474028157, 8.163316362729138706568539141366, 8.781250904515375357787719940940, 9.818642012087489242936422310128

Graph of the ZZ-function along the critical line